Fundamentals of Powder Diffractionand Structural Characterizationof Materials

Second Edition

Vitalij K. Pecharsky • Peter Y. Zavalij

Fundamentals of PowderDiffraction and StructuralCharacterizationof Materials

Second Edition

ABC

Vitalij K. PecharskyAnson Marston Distinguished Professor

of EngineeringDepartment of Materials Science

and Engineering and Ames Laboratoryof US Department of Energy

Iowa State UniversityAmes, Iowa 50011-3020USAvitkp@ameslab.gov

Peter Y. ZavalijDirector, X-Ray Crystallographic CenterDepartment of Chemistry and BiochemistryUniversity of MarylandCollege Park, Maryland 20742-4454USApzavalij@umd.edu

Cover illustration, created by Peter Zavalij, follows the book content. It is inspired by Salvador Dali’spainting “The Metamorphosis of Narcissus” where Narcissus (polycrystalline (Au,Ni)Sn4, courtesyLubov Zavalij) falls in love with his own reflection (diffraction pattern), transforms into an egg (reciprocallattice), and then into a flower (crystal structure in a physical space), which bears his name.

ISBN: 978-0-387-09578-3 e-ISBN: 978-0-387-09579-0DOI: 10.1007/978-0-387-09579-0

Library of Congress Control Number: 2008930122

c© Springer Science+Business Media, LLC 2009All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights

Printed on acid-free paper

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To those whose tracks we have followedand those who will follow ours

Note to Readers

Supplementary files, including color figures, powder diffraction data, examples, andweb links, can be found at www.springer.com/978-0-387-09578-3.

Note to Instructors

A solutions manual is available to instructors at www.springer.com/978-0-387-09578-3. Instructors must register to access these files.

Preface

A little over five years have passed since the first edition of this book appearedin print. Seems like an instant but also eternity, especially considering numerousdevelopments in the hardware and software that have made it from the laboratorytest beds into the real world of powder diffraction. This prompted a revision, whichhad to be beyond cosmetic limits. The book was, and remains focused on standardlaboratory powder diffractometry. It is still meant to be used as a text for teachingstudents about the capabilities and limitations of the powder diffraction method. Wealso hope that it goes beyond a simple text, and therefore, is useful as a reference topractitioners of the technique.

The original book had seven long chapters that may have made its use as a text in-convenient. So the second edition is broken down into 25 shorter chapters. The firstfifteen are concerned with the fundamentals of powder diffraction, which makes itmuch more logical, considering a typical 16-week long semester. The last ten chap-ters are concerned with practical examples of structure solution and refinement,which were preserved from the first edition and expanded by another example –solving the crystal structure of Tylenol R©.

Major revisions include an expanded discussion of nonconventional crystallo-graphic symmetry in Chap. 5, a short description of two new types of detectors thatare becoming common in laboratory powder diffractometry – real-time multiplestrip and multi wire detectors in Chap. 6, a brief introduction to the total scatteringanalysis in Chap. 10, a short section in Chap. 11 describing nonambient powderdiffractometry, an expanded discussion of quantitative phase analysis, including thebasics of how to quantify amorphous component in Chap. 13, an update about therecent advancements in the ab initio indexing, together with an example of a dif-ficult pseudo-symmetric case represented by Li[B(C2O4)2], and a major update ofChap. 15 dedicated to the fundamentals of Rietveld analysis, including a brief intro-duction of the mechanism of restraints, constraints, and rigid bodies. The collectionof problems that may be used by instructors to assess students’ progress and as self-exercises has also expanded. All problems related to the solution and refinement ofcrystal structures from powder diffraction data are assembled at the end of Chap. 25.

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Considering all these additions, something had to go. A major deletion from theearlier paper version is the section on X-ray safety, which has been moved to theelectronic part of the book. Readers familiar with the first edition know that the bookincluded a CD with electronic figures, experimental data, and solutions of all prob-lems. Over the years, both the publisher and we have had numerous inquiries frompeople who accidentally used the CD as a coaster, clay pigeon, or simply sat on itbefore making a backup copy. While each and every request about sending a copy ofthe CD was fulfilled, we thought that it makes more sense to have the electronic filesavailable online. The files are hosted by Springer (http://www.springer.com/978-0-387-09578-3) and they are made available to everyone who has the book. The filesinclude color figures, powder diffraction data, examples, web links, and solutions toall the problems found throughout the book. Files with the solutions of the problemsare only available to instructors, who must register with the publisher.

Finally, we would like to thank everyone who provided critique and feedback.Most important, we thank the readers who opted to buy our book with their hard-earned money thus providing enough votes for the publisher to consider this second,revised edition. It is our hope that this edition is met with even better acceptanceby our readers of students, practitioners, and instructors of the truly basic materialscharacterization technique, which is the powder diffraction method.

Ames, Iowa, October 2008 Vitalij K. PecharskyCollege Park, Maryland, October 2008 Peter Y. Zavalij

Preface to the First Edition

Without a doubt, crystals such as diamonds, emeralds and rubies, whose beauty hasbeen exposed by jewelry-makers for centuries, are enjoyed by everybody for theirperfect shapes and astonishing range of colors. Far fewer people take pleasure inthe internal harmony – atomic structure – which defines shapes and other proper-ties of crystals but remains invisible to the naked eye. Ordered atomic structures arepresent in a variety of common materials, for example, metals, sand, rocks or ice,in addition to the easily recognizable precious stones. The former usually consistof many tiny crystals and therefore, are called polycrystals, for example metals andice, or powders, such as sand and snow. Besides external shapes and internal struc-tures, the beauty of crystals can be appreciated from an infinite number of distinctdiffraction patterns they form upon interaction with certain types of waves, for ex-ample, X-rays. Similarly, the beauty of the sea is largely defined by a continuouslychanging but distinctive patterns formed by waves on the water’s surface.

Diffraction patterns from powders are recorded as numerical functions of a sin-gle independent variable, the Bragg angle, and they are striking in their fundamen-tal simplicity. Yet, a well-executed experiment encompasses an extraordinarily richvariety of structural information, which is encoded in a material- and instrument-specific distribution of the intensity of coherently scattered monochromatic waveswhose wavelengths are commensurate with lattice spacing. The utility of the pow-der diffraction method – one of the most essential tools in the structural character-ization of materials – has been tested for over 90 years of successful use in bothacademia and industry. A broad range of general-purpose and specialized powderdiffractometers are commonly available today, and just about every research projectthat involves polycrystalline solids inevitably begins with collecting a powder dif-fraction pattern. The pattern is then examined to establish or verify phase composi-tion, purity, and the structure of the newly prepared material. In fact, at least a basicidentification by employing powder diffraction data as a fingerprint of a substance,coupled with search-and-match among hundreds of thousands of known powderdiffraction patterns stored in various databases, is an unwritten mandate for everyserious work that involves crystalline matter.

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Throughout the long history of the technique, its emphasis underwent severalevolutionary and revolutionary transformations. Remarkably, the new developmentshave neither taken away, nor diminished the value of earlier applications of thepowder diffraction method; on the contrary, they enhanced and made them moreprecise and dependable. A noteworthy example is phase identification from powder-diffraction data, which dates back to the late 1930s (Hanawalt, Rinn, and Frevel).Over the years, this application evolved into the Powder Diffraction File

TMcontain-

ing reliable patterns of some 300,000 crystalline materials in a readily searchabledatabase format (Powder Diffraction File is maintained and distributed by the Inter-national Centre for Diffraction Data, http://www.icdd.com).

As it often happens in science and engineering, certain innovations may go unno-ticed for some time but when a critical mass is reached or exceeded, they stimulateunprecedented growth and expansion, never thought possible in the past. Both thesignificance and applications of the powder diffraction method have been drasticallyaffected by several directly related as well as seemingly unrelated developments thathave occurred in the recent past. First was the widespread transition from analogue(X-ray film) to digital (point, line, and area detectors) recording of scattered inten-sity, which resulted in the improved precision and resolution of the data. Second wasthe groundbreaking work by Rietveld, Young and many others, who showed that fullprofile powder diffraction data may be directly employed in structure refinement andsolution. Third was the availability of personal computers, which not only functionas instrument controllers, but also provide the much needed and readily availablecomputing power. Computers thus enable the processing of large arrays of data col-lected in an average powder diffraction experiment. Fourth was the invention andrapid evolution of the internet, which puts a variety of excellent, thoroughly testedcomputer codes at everyone’s fingertips, thanks to the visionary efforts of manybright and dedicated crystallographers.

Collectively, these major developments resulted in the revolutionary changes andopened new horizons for the powder diffraction technique. Not so long ago, if youwanted to establish the crystal structure of a material at the atomic resolution, vir-tually the only reliable choice was to grow an appropriate quality single crystal.Only then could one proceed with the collection of diffraction data from the crys-tal followed by a suitable data processing to solve the structure and refine relevantstructural parameters. A common misconception among the majority of crystallo-graphers was that powder diffraction has a well-defined niche, which is limited tophase identification and precise determination of unit cell dimensions. Over the pastten to twenty years the playing field has changed dramatically, and the ab initiostructure determination from powder diffraction data is now a reality. This raises thebar and offers no excuse for those who sidestep the opportunity to establish detailsof the distribution of atoms in the crystal lattice of every polycrystalline material,whose properties are under examination. Indeed, accurate structural knowledge ob-tained from polycrystals is now within reach. We believe that it will eventually leadto a much better understanding of structure-property relationships, which are criticalfor future advancements in materials science, chemistry, physics, natural sciences,and engineering.

Preface to the First Edition xi

Before a brief summary recounting the subject of this book, we are obliged tomention that our work was not conducted in a vacuum. Excellent texts describingthe powder diffraction method have been written, published, and used by the gener-ations of professors teaching the subject and by the generations of students learningthe trade in the past. Traditional applications of the technique have been excep-tionally well-covered by Klug and Alexander (1954), Azaroff and Buerger (1958),Lipson and Steeple (1970), Cullity (1956 and 1978), Jenkins and Snyder (1996),and Cullity and Stock (2001). There has never been a lack of reports describing themodern capabilities of powder diffraction, and they remain abundant in technicalliterature (Journal of Applied Crystallography, Acta Crystallographica, Powder Dif-fraction, Rigaku Journal, and others). A collective monograph, dedicated entirely tothe Rietveld method, was edited by Young and published in 1993. A second col-lection of reviews, describing the state of the art in structure determination frompowder diffraction data, appeared in 2002, and it was edited by David, Shankland,McCusker, and Baerlocher. These two outstanding and highly professional mono-graphs are a part of the multiple-volume series sponsored by the International Unionof Crystallography, and are solid indicators that the powder diffraction method hasbeen indeed transformed into a powerful and precise, yet readily accessible, struc-ture determination tool. We highly recommend all the books mentioned in this para-graph as additional reading to everyone, although the older editions are out of print.

Our primary motivation for this work was the absence of a suitable text thatcan be used by both the undergraduate and graduate students interested in pursu-ing in-depth knowledge and gaining practical experience in the application of thepowder diffraction method to structure solution and refinement. Here, we place em-phasis on powder diffraction data collected using conventional X-ray sources andgeneral-purpose powder diffractometers, which remain primary tools for thousandsof researchers and students in their daily experimental work. Brilliant synchrotronand powerful neutron sources, which are currently operational or in the process ofbecoming so around the world, are only briefly mentioned. Both may, and oftendo provide unique experimental data, which are out-of-reach for conventional pow-der diffraction especially when high pressure, high and low temperature, and otherextreme environments are of concern. The truth, however, is that the beam time isprecious, and both synchrotron and neutron sources are unlikely to become availableto everyone on a daily basis. Moreover, diffraction fundamentals remain the same,regardless of the nature of the employed radiation and the brilliance of the source.

This book has spawned from our affection and lasting involvement with the tech-nique, which began long ago in a different country, when both of us were workingour way through the undergraduate and then graduate programs in Inorganic Chem-istry at L’viv State University, one of the oldest and finest institutions of higher edu-cation in Ukraine. As we moved along, powder diffraction has always remained ontop of our research and teaching engagements. The major emphasis of our researchis to obtain a better understanding of the structure–property relationships of crys-talline materials, and both of us teach graduate-level powder diffraction courses atour respective departments – Materials Science and Engineering at Iowa State Uni-versity and Chemistry at the State University of New York (SUNY) at Binghamton.

xii Preface to the First Edition

Even before we started talking about this book, we were unanimous in our goals:the syllabi of two different courses were independently designed to be useful for anybackground, including materials science, solid-state chemistry, physics, mineralogy,and literally any other area of science and engineering, where structural informationat the atomic resolution is in demand. This philosophy, we hope, resulted in a textthat requires no prior knowledge of the subject. Readers are expected to have a gen-eral scientific and mathematical background of the order of the first two years of atypical liberal arts and sciences or engineering college.

The book is divided into seven chapters. The first chapter deals with essentialconcepts of crystallographic symmetry, which are intended to facilitate both the un-derstanding and appreciation of crystal structures. This chapter will also prepare thereader for the realization of the capabilities and limitations of the powder diffrac-tion method. It begins with the well-established notions of the three-dimensionalperiodicity of crystal lattices and conventional crystallographic symmetry. It endswith a brief introduction to the relatively young subject – the symmetry of aperiodiccrystals. Properties and interactions of symmetry elements, including examinationof both point and space groups, the concept of reciprocal space, which is employedto represent diffraction from crystalline solids, and the formal algebraic treatmentof crystallographic symmetry are introduced and discussed to the extent needed inthe context of the book.

The second chapter is dedicated to properties and sources of radiation suitablefor powder diffraction analysis, and gives an overview of the kinematical theoryof diffraction along with its consequences in structure determination. Here, readerslearn that the diffraction pattern of a crystal is a transformation of an ordered atomicstructure into a reciprocal space rather than a direct image of the former. Diffractionfrom crystalline matter, specifically from polycrystalline materials is described asa function of crystal symmetry, atomic structure, and conditions of the experiment.The chapter ends with a general introduction to numerical techniques enabling therestoration of the three-dimensional distribution of atoms in a lattice by the trans-formation of the diffraction pattern back into direct space.

The third chapter begins with a brief historical overview describing the powderdiffraction method and explains the principles, similarities, and differences amongthe variety of powder diffractometers available today. Since ionizing radiation andhighly penetrating and energetic particles are employed in powder diffraction, safetyis always a primary concern. Basic safety issues are concisely spelled out usingpolicies and procedures established at the US DOE’s Ames Laboratory as a prac-tical example. Sample preparation and proper selection of experimental conditionsare exceedingly important in the successful implementation of the technique. There-fore, the remainder of this chapter is dedicated to a variety of issues associated withspecimen preparation, data collection, and analysis of most common systematic er-rors that have an impact on every powder diffraction experiment.

Beginning from chapter four, key issues that arise during the interpretation ofpowder diffraction data, eventually leading to structure determination, are con-sidered in detail and illustrated by a variety of practical examples. This chapterdescribes preliminary processing of experimental data, which is critical in both

Preface to the First Edition xiii

qualitative and quantitative phase analyses. In addition to a brief overview of phaseidentification techniques and quantitative analysis, readers will learn how to deter-mine both the integrated intensities and angles of the observed Bragg peaks with thehighest achievable precision.

Chapter five deals with the first major hurdle, which is encountered in powderdiffraction analysis: unavoidably, the determination of any crystal structure startsfrom finding the shape, symmetry, and dimensions of the unit cell of the crystal lat-tice. In powder diffraction, finding the true unit cell from first principles may presentconsiderable difficulty because experimental data are a one-dimensional projectionof the three-dimensional reciprocal lattice. This chapter, therefore, introduces thereader to a variety of numerical techniques that result in the determination of preciseunit cell dimensions. The theoretical background is followed by multiple practicalexamples with varying complexity.

Chapter six is dedicated to the solution of materials’ structures, that is, here welearn how to find the distribution of atoms in the unit cell and create a completeor partial model of the crystal structure. The problem is generally far from trivial,and many structure solution cases in powder diffraction remain unique. Althoughstructure determination from powder data is not a wide-open and straight highway,knowing where to enter, how to proceed, and where and when to exit is equally vital.Hence, in this chapter both direct and reciprocal space approaches and some practicalapplications of the theory of kinematical diffraction to solving crystal structuresfrom powder data are explained and broadly illustrated. Practical examples start fromsimple, nearly transparent cases, and end with quite complex inorganic structures.

The solution of a crystal structure is considered complete only when multipleprofile variables and crystallographic parameters of a model have been fully refinedagainst the observed powder diffraction data. Thus, the last, the seventh chapterof this book describes the refinement technique, most commonly employed today,which is based on the idea suggested in the middle 1960s by Rietveld. Successfulpractical use of the Rietveld method, though directly related to the quality of pow-der diffraction data (the higher the quality, the more reliable the outcome), largelydepends on the experience and the ability of the user to properly select a sequencein which various groups of parameters are refined. In this chapter, we introducethe basic theory of Rietveld’s approach, followed by a series of hands-on exam-ples that demonstrate the refinement of crystal structures with various degrees ofcompleteness and complexity, models of which were partially or completely built inchapter six.

The book is supplemented by an electronic volume – compact disk – containingpowder diffraction data collected from a variety of materials that are used as exam-ples and in the problems offered at the end of every chapter. In addition, electronicversions of some 330 illustrations found throughout the book are also on the CD.Electronic illustrations, which we hope is useful to both instructors and students be-cause electronic figures are in color, are located in a separate folder /Figures on theCD. Three additional folders named /Problems, /Examples and /Solutions containexperimental data, which are required for solving problems, as self-exercises, andour solutions to the problems, respectively. The disk is organized as a web page,

xiv Preface to the First Edition

which makes it easy to navigate. All web links found in the book, are included onthe CD and can be followed by simply clicking on them. Every link is current asof January 2003. The compact disk is accessible using both Mac’s and PC’s, andpotential incompatibility problems have been avoided by using portable document,HTML, and ASCII formats.

Many people have contributed in a variety of ways in the making of this book.Our appreciation and respect goes to all authors of books, monographs, researcharticles, websites, and computer programs cited and used as examples through-out this text. We are indebted to our colleagues, Professor Karl Gschneidner, Jr.from Iowa State University, Professor Scott R.J. Oliver from SUNY at Bingham-ton, Professor Alexander Tishin from Moscow State University, Dr. Aaron Holmfrom Iowa State University, and Dr. Alexandra (Sasha) Pecharsky from Iowa StateUniversity, who read the entire manuscript and whose helpful advice and friendlycriticism made this book better. It also underwent a common-sense test, thanks toLubov Zavalij and Vitalij Pecharsky, Jr. Some of the experimental data and samplesused as the examples have been provided by Dr. Lev Akselrud from L’viv StateUniversity, Dr. Oksana Zaharko from Paul Scherrer Institute, Dr. Iver Anderson,Dr. Matthew Kramer, and Dr. John Snyder (all from Ames Laboratory, Iowa StateUniversity), and we are grateful to all of them for their willingness to share theresults of their unpublished work. Special thanks are in order to Professor KarlGschneidner, Jr. (Iowa State University) and Professor M. Stanley Whittingham(SUNY at Binghamton), whose perpetual attention and encouragement during ourwork on this book have been invaluable. Finally yet significantly, we extend ourgratitude to our spouses, Alexandra (Sasha) Pecharsky and Lubov Zavalij, and toour children, Vitalij Jr., Nadya, Christina, Solomia, and Martha, who handled ourvirtual absence for countless evenings and weekends with exceptional patience andunderstanding.

Ames, Iowa, January 2003 Vitalij K. PecharskyBinghamton, New York, January 2003 Peter Y. Zavalij

Contents

1 Fundamentals of Crystalline State and Crystal Lattice . . . . . . . . . . . . . . 11.1 Crystalline State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Crystal Lattice and Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Shape of the Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Crystallographic Planes, Directions, and Indices . . . . . . . . . . . . . . . . 8

1.4.1 Crystallographic Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Crystallographic Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Finite Symmetry Elements and Crystallographic Point Groups . . . . . . 172.1 Content of the Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Asymmetric Part of the Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Symmetry Operations and Symmetry Elements . . . . . . . . . . . . . . . . . 192.4 Finite Symmetry Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Onefold Rotation Axis and Center of Inversion . . . . . . . . . . 252.4.2 Twofold Rotation Axis and Mirror Plane . . . . . . . . . . . . . . . 262.4.3 Threefold Rotation Axis and Threefold Inversion Axis . . . 262.4.4 Fourfold Rotation Axis and Fourfold Inversion Axis . . . . . 272.4.5 Sixfold Rotation Axis and Sixfold Inversion Axis . . . . . . . 28

2.5 Interaction of Symmetry Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.1 Generalization of Interactions Between Finite

Symmetry Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.2 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 Fundamentals of Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Crystal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.9 Crystallographic Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.10 Laue Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.11 Selection of a Unit Cell and Bravais Lattices . . . . . . . . . . . . . . . . . . . 41

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2.12 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Infinite Symmetry Elements and Crystallographic Space Groups . . . . 513.1 Glide Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Screw Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Interaction of Infinite Symmetry Elements . . . . . . . . . . . . . . . . . . . . . 543.4 Crystallographic Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.1 Relationships Between Point Groups and Space Groups . . 573.4.2 Full International Symbols of Crystallographic Space

Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4.3 Visualization of Space-Group Symmetry

in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.4 Space Groups in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 International Tables for Crystallography . . . . . . . . . . . . . . . . . . . . . . . 633.6 Equivalent Positions (Sites) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6.1 General and Special Equivalent Positions . . . . . . . . . . . . . . 703.6.2 Special Sites with Points Located on Mirror Planes . . . . . . 713.6.3 Special Sites with Points Located on Rotation

and Inversions Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.6.4 Special Sites with Points Located on Centers

of Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.7 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Formalization of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1 Symbolic Representation of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.1 Finite Symmetry Operations . . . . . . . . . . . . . . . . . . . . . . . . . 774.1.2 Infinite Symmetry Operations . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Algebraic Treatment of Symmetry Operations . . . . . . . . . . . . . . . . . . 794.2.1 Transformation of Coordinates of a Point . . . . . . . . . . . . . . 794.2.2 Rotational Transformations of Vectors . . . . . . . . . . . . . . . . . 834.2.3 Translational Transformations of Vectors . . . . . . . . . . . . . . . 844.2.4 Combined Symmetrical Transformations of Vectors . . . . . . 854.2.5 Augmentation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.6 Algebraic Representation of Crystallographic

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2.7 Interaction of Symmetry Operations . . . . . . . . . . . . . . . . . . . 88

4.3 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Nonconventional Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.1 Commensurate Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2 Incommensurate Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Composite Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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5.4 Symmetry of Modulated Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.5 Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.6 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Properties, Sources, and Detection of Radiation . . . . . . . . . . . . . . . . . . . . 1076.1 Nature of X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Production of X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2.1 Conventional Sealed X-Ray Sources . . . . . . . . . . . . . . . . . . 1116.2.2 Continuous and Characteristic X-Ray Spectra . . . . . . . . . . . 1136.2.3 Rotating Anode X-Ray Sources . . . . . . . . . . . . . . . . . . . . . . 1166.2.4 Synchrotron Radiation Sources . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Other Types of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.4 Detection of X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.4.1 Detector Efficiency, Linearity, Proportionalityand Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.4.2 Classification of Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4.3 Point Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.4.4 Line and Area Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Fundamentals of Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.1 Scattering by Electrons, Atoms and Lattices . . . . . . . . . . . . . . . . . . . . 134

7.1.1 Scattering by Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.1.2 Scattering by Atoms and Atomic Scattering Factor . . . . . . 1387.1.3 Scattering by Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.2 Geometry of Diffraction by Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.2.1 Laue Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.2.2 Braggs’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.2.3 Reciprocal Lattice and Ewald’s Sphere . . . . . . . . . . . . . . . . 144

7.3 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 The Powder Diffraction Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.1 Origin of the Powder Diffraction Pattern . . . . . . . . . . . . . . . . . . . . . . . 1528.2 Representation of Powder Diffraction Patterns . . . . . . . . . . . . . . . . . . 1578.3 Understanding of Powder Diffraction Patterns . . . . . . . . . . . . . . . . . . 1598.4 Positions of Powder Diffraction Peaks . . . . . . . . . . . . . . . . . . . . . . . . 162

8.4.1 Peak Positions as a Function of Unit Cell Dimensions . . . . 1638.4.2 Other Factors Affecting Peak Positions . . . . . . . . . . . . . . . . 165

8.5 Shapes of Powder Diffraction Peaks . . . . . . . . . . . . . . . . . . . . . . . . . . 1688.5.1 Peak-Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.5.2 Peak Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.6 Intensity of Powder Diffraction Peaks . . . . . . . . . . . . . . . . . . . . . . . . . 182

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8.6.1 Integrated Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828.6.2 Scale Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.6.3 Multiplicity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.6.4 Lorentz-Polarization Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 1878.6.5 Absorption Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.6.6 Preferred Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1948.6.7 Extinction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8.7 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9 Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039.1 Structure Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.1.1 Population Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049.1.2 Temperature Factor (Atomic Displacement Factor) . . . . . . 2069.1.3 Atomic Scattering Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.1.4 Phase Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

9.2 Effects of Symmetry on the Structure Amplitude . . . . . . . . . . . . . . . . 2179.2.1 Friedel Pairs and Friedel’s Law . . . . . . . . . . . . . . . . . . . . . . . 2189.2.2 Friedel’s Law and Multiplicity Factor . . . . . . . . . . . . . . . . . . 220

9.3 Systematic Absences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.3.1 Lattice Centering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.3.2 Glide Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229.3.3 Screw Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

9.4 Space Groups and Systematic Absences . . . . . . . . . . . . . . . . . . . . . . . 2259.5 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

10 Solving the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.1 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.2 Phase Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

10.2.1 Patterson Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24610.2.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.2.3 Structure Solution from Powder Diffraction Data . . . . . . . . 253

10.3 Total Scattering Analysis Using Pair Distribution Function . . . . . . . 25510.4 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

11 Powder Diffractometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311.1 Brief History of the Powder Diffraction Method . . . . . . . . . . . . . . . . 26411.2 Beam Conditioning in Powder Diffractometry . . . . . . . . . . . . . . . . . . 269

11.2.1 Collimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27111.2.2 Monochromatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

11.3 Principles of Goniometer Design in Powder Diffractometry . . . . . . . 28011.3.1 Goniostats with Strip and Point Detectors . . . . . . . . . . . . . . 28311.3.2 Goniostats with Area Detectors . . . . . . . . . . . . . . . . . . . . . . . 287

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11.4 Nonambient Powder Diffractometry . . . . . . . . . . . . . . . . . . . . . . . . . . 29211.4.1 Variable Temperature Powder Diffractometry . . . . . . . . . . . 29211.4.2 Principles of Variable Pressure Powder Diffractometry . . . 29411.4.3 Powder Diffractometry in High Magnetic Fields . . . . . . . . . 296

11.5 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29911.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

12 Collecting Quality Powder Diffraction Data . . . . . . . . . . . . . . . . . . . . . . . 30112.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

12.1.1 Powder Requirements and Powder Preparation . . . . . . . . . . 30112.1.2 Powder Mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30412.1.3 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31012.1.4 Sample Thickness and Uniformity . . . . . . . . . . . . . . . . . . . . 31112.1.5 Sample Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31312.1.6 Effects of Sample Preparation on Powder

Diffraction Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31412.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

12.2.1 Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31812.2.2 Monochromatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32012.2.3 Incident Beam Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32212.2.4 Diffracted Beam Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . 32512.2.5 Variable Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32912.2.6 Power Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33012.2.7 Classification of Powder Diffraction Experiments . . . . . . . 33112.2.8 Step Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33112.2.9 Continuous Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33412.2.10 Scan Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

12.3 Quality of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33812.3.1 Quality of Intensity Measurements . . . . . . . . . . . . . . . . . . . . 33912.3.2 Factors Affecting Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 342

12.4 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34312.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

13 Preliminary Data Processing and Phase Analysis . . . . . . . . . . . . . . . . . . . 34713.1 Interpretation of Powder Diffraction Data . . . . . . . . . . . . . . . . . . . . . . 34813.2 Preliminary Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

13.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35513.2.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35913.2.3 Kα2 Stripping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36113.2.4 Peak Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36313.2.5 Profile Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

13.3 Phase Identification and Quantitative Analysis . . . . . . . . . . . . . . . . . . 37713.3.1 Crystallographic Databases . . . . . . . . . . . . . . . . . . . . . . . . . . 37713.3.2 Phase Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38213.3.3 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

xx Contents

13.3.4 Phase Contents from Rietveld Refinement . . . . . . . . . . . . . . 39413.3.5 Determination of Amorphous Content or Degree

of Crystallinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39513.4 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39913.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

14 Determination and Refinement of the Unit Cell . . . . . . . . . . . . . . . . . . . . 40714.1 The Indexing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40714.2 Known Versus Unknown Unit Cell Dimensions . . . . . . . . . . . . . . . . . 41014.3 Indexing: Known Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

14.3.1 High Symmetry Indexing Example . . . . . . . . . . . . . . . . . . . . 41414.3.2 Other Crystal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

14.4 Reliability of Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42114.4.1 The FN Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42414.4.2 The M20(MN) Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . 425

14.5 Introduction to Ab Initio Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42614.6 Cubic Crystal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

14.6.1 Primitive Cubic Unit Cell: LaB6 . . . . . . . . . . . . . . . . . . . . . . 43014.6.2 Body-Centered Cubic Unit Cell: U3Ni6Si2 . . . . . . . . . . . . . 432

14.7 Tetragonal and Hexagonal Crystal Systems . . . . . . . . . . . . . . . . . . . . 43414.7.1 Indexing Example: LaNi4.85 Sn0.15 . . . . . . . . . . . . . . . . . . . . 437

14.8 Automatic Ab Initio Indexing Algorithms . . . . . . . . . . . . . . . . . . . . . 44014.8.1 Indexing in Direct Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44114.8.2 Indexing in Reciprocal Space . . . . . . . . . . . . . . . . . . . . . . . . 444

14.9 Unit Cell Reduction Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44714.9.1 Delaunay–Ito Transformation . . . . . . . . . . . . . . . . . . . . . . . . 44814.9.2 Niggli Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

14.10 Automatic Ab Initio Indexing: Computer Codes . . . . . . . . . . . . . . . . 45014.10.1 TREOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45114.10.2 DICVOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45314.10.3 ITO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45414.10.4 Selecting a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

14.11 Ab Initio Indexing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45714.11.1 Hexagonal Indexing: LaNi4.85Sn0.15 . . . . . . . . . . . . . . . . . . . 45714.11.2 Monoclinic Indexing: (CH3NH3)2Mo7O22 . . . . . . . . . . . . . 46214.11.3 Triclinic Indexing: Fe7(PO4)6 . . . . . . . . . . . . . . . . . . . . . . . . 46614.11.4 Pseudo-Hexagonal Indexing: LiB(C2O4)2 . . . . . . . . . . . . . . 470

14.12 Precise Lattice Parameters and Linear Least Squares . . . . . . . . . . . . 47314.12.1 Linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47514.12.2 Precise Lattice Parameters from Linear

Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47714.13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48514.14 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48514.15 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

Contents xxi

15 Solving Crystal Structure from Powder Diffraction Data . . . . . . . . . . . . 49715.1 Ab Initio Methods of Structure Solution . . . . . . . . . . . . . . . . . . . . . . . 497

15.1.1 Conventional Reciprocal Space Methods . . . . . . . . . . . . . . . 49815.1.2 Conventional Direct Space Modeling . . . . . . . . . . . . . . . . . . 49915.1.3 Unconventional Direct, Reciprocal, and Dual Space

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50015.1.4 Validation and Completion of the Model . . . . . . . . . . . . . . . 505

15.2 The Content of the Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50615.3 Pearson’s Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50915.4 Finding Structure Factors from Powder Diffraction Data . . . . . . . . . 51015.5 Nonlinear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51315.6 Quality of Profile Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

15.6.1 Visual Assessment of the Quality of Profile Fitting . . . . . . 51815.6.2 Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

15.7 The Rietveld Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52415.7.1 Fundamentals of the Rietveld Method . . . . . . . . . . . . . . . . . 52715.7.2 Classes of Rietveld Refinement Parameters . . . . . . . . . . . . . 52915.7.3 Restraints, Constraints, and Rigid-Bodies . . . . . . . . . . . . . . 53115.7.4 Figures of Merit and Quality of Rietveld Refinement . . . . . 53815.7.5 Common Problems and How to Deal with Them . . . . . . . . 53915.7.6 Termination of Rietveld Refinement . . . . . . . . . . . . . . . . . . . 542

15.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54315.9 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

16 Crystal Structure of LaNi4.85Sn0.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54716.1 Full Pattern Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54916.2 Solving the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55616.3 Rietveld Refinement Using Cu Kα1,2 Radiation . . . . . . . . . . . . . . . . . 560

16.3.1 Scale Factor and Profile Parameters . . . . . . . . . . . . . . . . . . . 56116.3.2 Overall Atomic Displacement Parameter . . . . . . . . . . . . . . . 56316.3.3 Individual Parameters, Free and Constrained Variables . . . 56416.3.4 Anisotropic Atomic Displacement Parameters . . . . . . . . . . 56716.3.5 Multiple Phase Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 56716.3.6 Refinement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

16.4 Rietveld Refinement Using Mo Kα1,2 Radiation . . . . . . . . . . . . . . . . 56916.5 Combined Refinement Using Different Sets of Diffraction Data . . . 573

17 Crystal Structure of CeRhGe3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57917.1 Full Pattern Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57917.2 Solving the Crystal Structure from X-Ray Data . . . . . . . . . . . . . . . . . 583

17.2.1 Highest Symmetry Attempt . . . . . . . . . . . . . . . . . . . . . . . . . . 58417.2.2 Low-Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

17.3 Solving the Crystal Structure from Neutron Data . . . . . . . . . . . . . . . . 58917.4 Rietveld Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

17.4.1 X-Ray Data, Correct Low Symmetry Model . . . . . . . . . . . . 595

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17.4.2 X-Ray Data, Wrong High-Symmetry Model . . . . . . . . . . . . 59817.4.3 Neutron Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

18 Crystal Structure of Nd5Si4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60318.1 Full Pattern Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60318.2 Solving the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60418.3 Rietveld Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

19 Empirical Methods of Solving Crystal Structures . . . . . . . . . . . . . . . . . . 61119.1 Crystal Structure of Gd5Ge4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61219.2 Crystal Structure of Gd5Si4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61519.3 Crystal Structure of Gd5Si2Ge2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61619.4 Rietveld Refinement of Gd5Ge4, Gd5Si4, and Gd5Si2Ge2 . . . . . . . . 620

19.4.1 Gd5Ge4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62019.4.2 Gd5Si4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62319.4.3 Gd5Si2Ge2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

19.5 Structure–Property Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

20 Crystal Structure of NiMnO2(OH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63320.1 Observed Structure Factors from Experimental Data . . . . . . . . . . . . . 63320.2 Solving the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63620.3 A Few Notes About Using GSAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64020.4 Completion of the Model and Rietveld Refinement . . . . . . . . . . . . . . 643

20.4.1 Initial Refinement Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64320.4.2 Where Is Mn and Where Is Ni? . . . . . . . . . . . . . . . . . . . . . . . 64720.4.3 Finalizing the Refinement of the Model Without

Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64820.4.4 Locating Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64820.4.5 Combined Rietveld Refinement . . . . . . . . . . . . . . . . . . . . . . . 650

21 Crystal Structure of tmaV3O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65521.1 Observed Structure Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65621.2 Solving the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65821.3 Completion of the Model and Rietveld Refinement . . . . . . . . . . . . . . 661

21.3.1 Unrestrained Rietveld Refinement . . . . . . . . . . . . . . . . . . . . 66221.3.2 Rietveld Refinement with Restraints . . . . . . . . . . . . . . . . . . . 665

22 Crystal Structure of ma2Mo7O22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66922.1 Possible Model of the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . 66922.2 Rietveld Refinement and Completion of the Model . . . . . . . . . . . . . . 672

23 Crystal Structure of Mn7(OH)3(VO4)4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67923.1 Solving the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68023.2 Rietveld Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68223.3 Determining Chemical Composition . . . . . . . . . . . . . . . . . . . . . . . . . . 685

Contents xxiii

24 Crystal Structure of FePO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69124.1 Building and Optimizing the Model of the Crystal Structure . . . . . . 69224.2 Rietveld Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

25 Crystal Structure of Acetaminophen, C8H9NO2 . . . . . . . . . . . . . . . . . . . . 70325.1 Ab Initio Indexing and Le Bail Fitting . . . . . . . . . . . . . . . . . . . . . . . . . 70525.2 Solving the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

25.2.1 Creating a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70925.2.2 Optimizing the Model (Solving the Structure) . . . . . . . . . . . 713

25.3 Restrained Rietveld Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71725.4 Chapters 15–25: Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 72125.5 Chapters 15–25: Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729